Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.
Reason (R) : Product of two diagonal matrices of same order is commutative.
Reason (R) : Product of two diagonal matrices of same order is commutative.
Show Hint
For A and R questions, always check if R is a specific example or a general underlying cause.
- Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
The Correct Option is B
Solution and Explanation
Concept:
Matrix multiplication is generally non-commutative, which means: \[ AB\neq BA \] for arbitrary square matrices \(A\) and \(B\).
However, some special matrices, such as diagonal matrices of the same order, commute under multiplication.
Step 1: Check Assertion (A)
Consider: \[ A= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \qquad B= \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \] Then: \[ AB= \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \] while: \[ BA= \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \] Therefore: \[ AB\neq BA \] Hence, Assertion (A) is true.
Step 2: Check Reason (R)
Let \(D_1\) and \(D_2\) be two diagonal matrices: \[ D_1= \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}, \qquad D_2= \begin{bmatrix} c & 0 \\ 0 & d \end{bmatrix} \] Then: \[ D_1D_2= \begin{bmatrix} ac & 0 \\ 0 & bd \end{bmatrix} \] and: \[ D_2D_1= \begin{bmatrix} ca & 0 \\ 0 & db \end{bmatrix} \] Since \(ac=ca\) and \(bd=db\): \[ D_1D_2=D_2D_1 \] Hence, Reason (R) is true.
Step 3: Check whether R explains A
Reason (R) describes a special case in which matrix multiplication is commutative.
It does not explain why matrix multiplication is generally non-commutative.
Final Answer:
\[ \boxed{\text{Both A and R are true, but R is not the correct explanation of A.}} \]
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Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.
Reason (R) : Product of two diagonal matrices of same order is commutative. - If \( A \) and \( B \) are square matrices of same order, then which of the following statements is/are always true?
(i) $(A+B)(A-B)=A^2-B^2$
(ii) $AB=BA$
(iii) $(A+B)^2=A^2+AB+BA+B^2$
(iv) $AB=0\Rightarrow A=0\text{ or }B=0$}
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